---
title: Tensor Dynamic Mode Decomposition (TDMD)
url: https://www.emergentmind.com/topics/tensor-dynamic-mode-decomposition-tdmd
type: topic
---

# Tensor Dynamic Mode Decomposition (TDMD)

Tensor Dynamic Mode Decomposition (TDMD) denotes tensorial extensions of dynamic mode decomposition for spatiotemporal data that are naturally represented as tensors rather than as flattened matrices. In the 2025 third-order T-product formulation, TDMD identifies a third-order transition tensor \(\mathscr A\) satisfying \(\mathscr X_{t+1}=\mathscr A\star\mathscr X_t\) and extracts tensorial dynamic modes and temporal dynamics from that operator; related literature also includes tensor-based DMD in Tensor-Train (TT) form and tensor-based DMD in the \(\star_M\)-product framework, all motivated by the observation that conventional DMD is matrix-based and may be inefficient or inadequate for images, videos, higher-order networks, and other inherently multidimensional data [2508.02627] [1606.06625] [2508.10126].

## 1. Position within dynamic mode decomposition

Classical DMD starts from two snapshot arrays,
\[
X_1=\bigl[x(0)\;\;x(\Delta t)\;\cdots\;x((M-1)\Delta t)\bigr],\qquad
X_2=\bigl[x(\Delta t)\;\;x(2\Delta t)\;\cdots\;x(M\Delta t)\bigr],
\]
and seeks a best-fit linear map \(A\) with
\[
A\,X_1\approx X_2,\qquad A=X_2X_1^+.
\]
After a truncated SVD \(X_1=U\Sigma V^*\), one forms the reduced operator
\[
\tilde A=U^*X_2V\Sigma^{-1},
\]
diagonalizes it, and reconstructs modes and temporal coefficients [2509.22177].

Tensor formulations retain that DMD logic while replacing matrix storage and matrix-only linear algebra by tensor representations, tensor pseudoinverses, and reduced operators defined in tensor algebras or compressed tensor networks. The literature contains several non-equivalent realizations of this idea.

| Formulation | Data representation | Reduced or fitted operator |
|---|---|---|
| Third-order T-product TDMD | \(\mathscr X_t\in\mathbb R^{n\times h\times m}\) | \(\mathscr A=\mathscr X_+\star\mathscr X_-^\dagger\) |
| TT-based tensor DMD | \(\mathcal X,\mathcal Y\in\mathbb R^{n_1\times\cdots\times n_d\times m}\) | \(\widetilde A=(M^TP)(QN^T)\Sigma^{-1}\) |
| \(\star_M\)-based tensor DMD | \(\mathcal X,\mathcal Y\in\mathbb C^{m\times p\times n}\) | \(\mathcal A_{\rm TDMD}=\mathcal Y\star_M\mathcal X^\dagger\) |

A common misconception is that TDMD names a single canonical algorithm. The literature instead presents distinct constructions based on different tensor models and operator algebras. This suggests that TDMD is best understood as a family of tensorial DMD schemes rather than a unique procedure.

## 2. Tensor models and algebraic foundations

The third-order T-product formulation is built on block-circulant unfolding and FFT-based slice-wise linear algebra. For \(\mathscr X\in\mathbb R^{n\times h\times m}\), the T-product with \(\mathscr Y\in\mathbb R^{h\times s\times m}\) is
\[
\mathscr X\star\mathscr Y
=
fold\!\bigl(bcirc(\mathscr X)\;unfold(\mathscr Y)\bigr)\in\mathbb R^{n\times s\times m}.
\]
The framework includes a T-identity, T-transpose, T-pseudoinverse, and T-orthogonality. Tensor SVD and tensor eigenvalue decomposition generalize matrix SVD and EVD:
\[
\mathscr X=\mathscr U\star\mathscr S\star\mathscr V^\top,\qquad
\mathscr A=\mathscr W\star\mathscr D\star\mathscr W^{-1},
\]
with \(\mathscr S\) and \(\mathscr D\) F-diagonal. Both TSVD and TEVD are implemented efficiently by applying FFT along the third mode, reducing to \(m\) independent matrix SVDs or EVDs [2508.02627].

The 2016 tensor-based DMD uses the Tensor-Train format. A \(d\)-way tensor \(\mathcal T\in\mathbb R^{n_1\times\cdots\times n_d}\) is represented as
\[
\mathcal T_{i_1,\dots,i_d}
=
\sum_{k_0=1}^{r_0}\sum_{k_1=1}^{r_1}\cdots\sum_{k_d=1}^{r_d}
T^{(1)}_{k_0,i_1,k_1}\cdots T^{(d)}_{k_{d-1},i_d,k_d},
\]
with TT-ranks \(r=(r_0,\dots,r_d)\), \(r_0=r_d=1\), and cores \(T^{(\mu)}\in\mathbb R^{r_{\mu-1}\times n_\mu\times r_\mu}\). The memory requirement is
\[
\sum_{\mu=1}^d r_{\mu-1}n_\mu r_\mu.
\]
The exposition explicitly attributes the TT format to Oseledets ’11, and under the standard successive-SVD construction the first \(d-1\) cores become left-orthonormal [1606.06625].

The \(\star_M\)-product framework defines tensor multiplication through a unitary transform \(M\in\mathbb C^{n\times n}\). For \(\mathcal X\in\mathbb C^{m\times p\times n}\), one first forms \(\widehat{\mathcal X}=\mathcal X\times_3 M\), multiplies frontal slices in the transform domain, and returns by \(\times_3 M^*\). The product is associative, has an identity tensor, and supports a slice-wise conjugate transpose, pseudoinverse, and \(\star_M\)-unitary tensors. The same exposition states, following Kilmer–Kilmer–Kernfeld ’21, that one obtains a full \(\star_M\)-SVD
\[
\mathcal A=\mathcal U\star_M\mathcal S\star_M\mathcal V^*,
\]
and that truncation to the largest \(k\) nonzero tubes gives the best Frobenius-norm t-rank-\(k\) approximation [2508.10126].

## 3. Algorithmic constructions

In the third-order T-product formulation, one starts from a sequence of tensor snapshots \(\{\mathscr X_t\}_{t=0}^T\), each in \(\mathbb R^{n\times h\times m}\), and forms
\[
\mathscr X_-=\bigl[\mathscr X_0\;\;\mathscr X_1\;\dots\;\mathscr X_{T-1}\bigr],\qquad
\mathscr X_+=\bigl[\mathscr X_1\;\;\mathscr X_2\;\dots\;\mathscr X_T\bigr].
\]
The goal is to identify \(\mathscr A\in\mathbb R^{n\times n\times m}\) such that
\[
\mathscr X_+=\mathscr A\star\mathscr X_-,
\]
and to extract tensorial modes \(\mathscr M\) and temporal dynamics \(\mathscr D\). The reduced-order construction is:
\[
\mathscr X_-=\mathscr U\star\mathscr S\star\mathscr V^\top,
\]
\[
\tilde{\mathscr A}
=
\mathscr U^\top\star\mathscr X_+\star\mathscr V\star\mathscr S^{-1},
\]
\[
\tilde{\mathscr A}
=
\mathscr W\star\mathscr D\star\mathscr W^{-1},
\qquad
\mathscr M=\mathscr U\star\mathscr W.
\]
The corresponding pseudocode performs FFT along mode 3, computes slice-wise economy SVDs, builds reduced matrices \(A_j\), diagonalizes them, and assembles \(\tilde{\mathscr A}\) and \(\mathscr M\) by inverse FFT and un-bcirc operations [2508.02627].

The TT-based construction begins with snapshots \(\mathbf x_i,\mathbf y_i\in\mathbb R^{n_1\times\cdots\times n_d}\), \(\mathbf y_i=F(\mathbf x_i)\), gathered into \((d+1)\)-mode tensors \(\mathcal X\) and \(\mathcal Y\). Standard DMD would vectorize these to \(X=\mathrm{unfold}(\mathcal X)\), \(Y=\mathrm{unfold}(\mathcal Y)\), then form \(A=YX^+\). The TT method instead computes the pseudoinverse of a tensor unfolding directly from TT cores. After left-orthonormalizing early cores, right-orthonormalizing later cores, and applying an SVD to the middle core, one obtains
\[
M=M_{\rm left}\Sigma M_{\rm right},\qquad
M^+=M_{\rm right}^{T}\Sigma^{-1}M_{\rm left}^{T}.
\]
This is the main theoretical result stated in the paper: no large-scale optimization is required, and the result is exact up to TT-truncation error. The reduced DMD matrix is then assembled from small factors,
\[
\widetilde A=M^T(PQ)(\Sigma N)^+=\bigl(M^TP\bigr)\bigl(QN^T\bigr)\Sigma^{-1},
\]
and standard or exact modes are recovered by replacing the final TT core by the relevant low-dimensional vector or coefficient block [1606.06625].

In the \(\star_M\)-product formulation, the tensor-regression viewpoint is explicit:
\[
\min_{\mathcal A}\|\mathcal Y-\mathcal A\star_M\mathcal X\|_F^2,
\qquad
\mathcal A_{\rm TDMD}=\mathcal Y\star_M\mathcal X^\dagger.
\]
After a truncated \(\star_M\)-SVD,
\[
\mathcal X\approx \mathcal U\star_M\mathcal S\star_M\mathcal V^*,
\]
one forms the reduced tensor
\[
\mathcal K
=
\mathcal U^*\star_M\mathcal Y\star_M\mathcal V\star_M\mathcal S^\dagger,
\]
computes a \(\star_M\)-Schur decomposition
\[
\mathcal K=\mathcal W\star_M\mathcal T\star_M\mathcal W^*,
\]
and defines the tensor of modes by
\[
\mathcal Z=\mathcal U\star_M\mathcal W.
\]
State reconstruction is then written as
\[
\mathcal X_t\approx \mathcal Z\star_M\mathcal T^t\star_M\mathcal G,
\qquad
\mathcal G=\mathcal Z^*\star_M\mathcal X_0.
\]
The same work also gives a randomized streaming algorithm that maintains two sketches, updates them batchwise with Gaussian sketch-tensors, and then invokes a static \(\star_M\)-DMD on the low-rank approximation [2508.10126].

A further extension integrates tensor-train DMD into a predictor-corrector method for nonequilibrium Green’s function calculations in quantics tensor-train form. There, \(X_1\) is orthogonalized and decomposed about a central bond as
\[
X_1=U\circ S\circ V^\dagger,\qquad X_1^+=VS^{-1}U^\dagger,\qquad
\tilde A=U^\dagger X_2VS^{-1}.
\]
After eigendecomposition, the predictor takes the form
\[
X_{\rm pred}=\Phi\circ\Lambda^m\circ b,
\]
and is coupled to a causality-preserving block-time-stepping corrector that iterates only on newly added time blocks [2509.22177].

## 4. Computational scaling and compression

The third-order T-product paper provides an explicit complexity comparison. Let \(n\) be the spatial dimension, \(T\) the number of snapshots, \(h\) the lateral mode size, \(m\) the number of tensor channels, and \(r=\min(n,T)\). TDMD slice-wise performs an SVD of an \(n\times T\) matrix per frontal slice, giving
\[
\text{Time}_{TDMD}
=
O\bigl(m[\,nT\min(n,T)+\log m\,(nT+hT)]\bigr).
\]
Standard flatten-then-DMD instead requires the SVD of an \((nm)\times(Th)\) matrix,
\[
\text{Time}_{DMD}
=
O\bigl(nm\;Th\;\min(nm,Th)\bigr)
=
O(n^2m^2T)\quad\text{assuming }nm>Th.
\]
Memory-wise, TDMD stores \(O(nrm)\) parameters in \(\mathscr U\) plus \(O(r^2m)\) in \(\tilde{\mathscr A}\), versus \(O(n^2m)\) for full DMD [2508.02627].

The TT formulation emphasizes that matrix DMD on ambient dimension \(N=\prod_{\mu=1}^dn_\mu\) requires \(\mathcal O(Nm)\) memory for \(X,Y\) and \(\mathcal O(N^2)\) if \(A\) is formed, while the TT representation uses
\[
\mathcal O\!\bigl(\sum_\mu r_{\mu-1}n_\mu r_\mu\bigr)
\]
memory and has a cost to build \(\widetilde A\) from small factors of \(\mathcal O(dr^3+r^2m)\). The paper’s conclusion is that if the data admits low TT ranks \(r_\mu\ll\min(N,m)\), one gains orders-of-magnitude in time and memory [1606.06625].

The \(\star_M\)-product work gives a parallel cost comparison. For matrix DMD on \(N=mn\) and \(p=T\), the dominant term is \(O(Np\min(N,p))\). For \(\star_M\)-DMD with t-rank \(k\), the t-SVD cost is
\[
O(mpn\min(m,p)+2mp\,T_M),
\]
the Schur step costs \(O(nk^3)\), and mode recombination costs \(O(nmk^2)\), with \(T_M=O(n\log n)\) if \(M\) is FFT-based. It also states that transforms and per-slice operations are embarrassingly parallel [2508.10126].

For QTT-based predictor-corrector TDMD, QTT-SVD and pseudoinverse for DMD cost \(\mathcal O(D^3R)\), contractions cost \(\mathcal O(D^4R)\), and the block solver per iteration is also \(\mathcal O(D^4R)\), where \(D\) is bond dimension and \(R=\log_2N_t\). The corresponding motivation is that direct matrix representations of two-time functions scale as \(\mathcal O(N_t^2)\), whereas QTT storage scales as \(\mathcal O(D^2R)\) [2509.22177].

Across these variants, the shared computational principle is low-rank tensor structure. What differs is the algebra in which “rank” is defined: tubal rank in the T-product setting, TT-ranks in TT/QTT, and t-rank in the \(\star_M\)-product framework.

## 5. Empirical performance and application domains

The 2025 third-order T-product TDMD paper reports two benchmark categories. On synthetic data consisting of \(10\times1\times6\) tensors with 20 snapshots, TDMD achieves relative reconstruction errors on the order of \(10^{-11}\) to \(10^{-6}\) across rank truncations \(r=10\) down to \(r=4\), whereas standard DMD with flattening gives a fixed error of approximately \(1.35\), insensitive to truncation. Runtime is approximately \(0.04\)–\(0.06\) s for TDMD versus \(0.09\)–\(0.20\) s for DMD, and memory usage is \(0.27\)–\(0.58\) MB versus \(25\)–\(55\) MB. On video data with \(60\times60\times20\) frames, TDMD yields per-frame relative error of approximately \(0.01\)–\(0.03\), while DMD yields approximately \(0.1\)–\(0.2\); in dynamic component separation, TDMD cleanly extracts a static background and moving square, while DMD mixing persists [2508.02627].

The 2016 TT-based study illustrates tensor DMD on fluid-dynamics problems. For a two-dimensional von Kármán vortex street on a \(60\times120\) grid with \(m=101\) snapshots, standard DMD uses \(X,Y\in\mathbb R^{7200\times100}\), whereas the tensor formulation stores \(\mathcal X,\mathcal Y\in\mathbb R^{60\times120\times100}\) in TT format with ranks \([1,60,r_2,1]\) and \(r_2=49\). The reported wall-clock runtime is approximately \(0.7\,\text{s}\) for standard DMD and \(0.05\,\text{s}\) for TDMD, corresponding to a \(14\times\) speed-up, and visual DMD modes match to machine precision. For two merging vortices with \(n\) up to \(1400\) and \(m=446\) time steps, the paper reports approximately \(7\times\) speed-up for exact TT and up to \(16\times\) for \(\varepsilon=10^{-10}\), with relative errors in eigenvalues and modes below \(10^{-13}\) for small \(\varepsilon\). For flow around a blunt body with data \(\mathcal X\in\mathbb R^{150\times85\times80\times1000}\), TT truncation with \(\varepsilon=0.05\) yields runtime \(52\,\text{s}\) versus \(125\,\text{s}\), first two eigenvalue error approximately \(10^{-4}\), and mode-error \(\lesssim10^{-2}\). The paper’s summary states that TDMD reduced memory and runtime by factors of \(5\)–\(20\) while producing the same dynamical spectra and spatial modes up to TT-truncation error [1606.06625].

The \(\star_M\)-product study evaluates cylinder-wake flow at \(\mathrm{Re}=100\), NOAA sea-surface temperature data, and a streaming cylinder dataset. On the cylinder-wake dataset of 150 snapshots of vorticity on a \(450\times200\) grid, \(\star\)-DMDII consistently attains lower reconstruction relative error at fixed storage, often half or less than either DMD or \(\star\)-DMD. On the first 200 of 1,727 weekly NOAA SST maps on a \(360\times180\) grid, standard DMD slightly outperforms \(\star\)-DMDII at low storage, but the errors converge as storage grows. In a streaming test with 200 snapshots on a \(640\times80\) grid split into 20 batches of 10, streaming \(\star\)-DMDII yields \(20\)–\(30\%\) lower reconstruction error on each batch and overall, and end-to-end CPU time for \(\star\)-DMD is often \(2\)–\(5\times\) faster at the same error level [2508.10126].

The QTT-NEGF predictor-corrector application extends tensor DMD into many-body nonequilibrium simulation. Applied to the Hubbard model on a \(32\times32\) lattice within the nonequilibrium \(GW\) approximation, it demonstrates stable propagation up to \(t_{\mathrm{max}}=512\) inverse hoppings, exceeding both matrix-based implementations and previous QTT solvers. The predictor typically reduces the initial error on a new block to approximately \(10^{-2}\), and the corrector then requires \(\mathcal O(10)\) iterations to reach approximately \(10^{-4}\) [2509.22177].

## 6. Implementation practice and conceptual boundaries

The third-order T-product paper gives explicit implementation guidelines. Tubal rank \(r\) is chosen by inspecting the singular-tube Frobenius norms \(\|\mathscr S_{jj:}\|\) and dropping those below a tolerance \(\tau\), with \(\tau=10^{-3}\) given as an example. For separating persistent versus transient modes, it recommends \(\epsilon\approx0.995\) on \(\|\mathscr D_{jj:}\|\). It also recommends the MATLAB “tensor-tensor-product-toolbox” available at `github.com/lu-group`, or Python implementations based on FFT with NumPy/SciPy for slice-wise SVD/EVD, together with three efficiency tips: precompute FFT along mode 3 once, use economy-size SVD with \(r\ll n,T\), and parallelize across slices \(j=1,\dots,m\) [2508.02627].

The \(\star_M\)-product paper places tensor DMD in direct relation to traditional DMD and physics-informed DMD. It states that
\[
\min_{\mathcal A}\|\mathcal Y-\mathcal A\star_M\mathcal X\|_F^2
\]
is equivalent to a constrained matrix regression over a structured subspace \(\mathcal A_M\) of block-structured matrices, and interprets \(\star\)-DMD as the case in which the operator must lie in the \(\star\)-algebra defined by \(M\). This provides an explicit connection to piDMD, which also imposes subspace constraints on \(A\) [2508.10126].

A second misconception is that tensorization by itself guarantees exact recovery or superior results. The literature qualifies that point. In the TT framework, the pseudoinverse theorem is exact apart from TT-truncation error [1606.06625]. In the third-order T-product formulation, the practical algorithm is a reduced-order approximation with chosen tubal rank [2508.02627]. In the \(\star_M\)-product setting, accuracy depends on the selected transform \(M\), the truncation strategy, and whether one uses fixed-rank tr-tSVDM or slice-adaptive tr-tSVDMII [2508.10126]. The numerical evidence consistently favors structure-preserving tensor formulations over flatten-then-DMD on the reported examples, but the reported comparisons are framework-specific rather than universal.

A third misconception is that tensor DMD is confined to fluid mechanics or image-like data. The cited applications do include vortex streets, merging vortices, blunt-body flow, cylinder wakes, sea-surface temperature fields, and video sequences, but the QTT predictor-corrector formulation is presented as a general framework not limited to nonequilibrium Green’s functions, requiring only that data at fixed times be stored in QTT form and that one can perform a self-consistent or nonlinear update on a small time block while holding the past fixed [2509.22177].

Taken together, these works define TDMD as a technically diverse but conceptually coherent research area: DMD-style Koopman-oriented regression and modal analysis carried out in tensor representations that preserve multiway structure, expose low-rank organization, and replace large dense matrix operations by tensor-algebraic or tensor-network computations.

Source: https://www.emergentmind.com/topics/tensor-dynamic-mode-decomposition-tdmd